Meromorphic Parabolic Starlike Functions Associated with -Hypergeometric Series

We introduce a new class of meromorphic parabolic starlike functions with a fixed point defined in the punctured unit disk involving the -hypergeometric functions. We obtained coefficient inequalities, growth and distortion inequalities, and closure results for functions . We further established some results concerning convolution and the partial sums. 1. Introduction Let be a fixed point in the unit disc . Denote by the class of functions which are regular and Also denote by , the subclass of consisting of the functions of the form which are analytic in . Note that is subclasses of consisting of univalent functions in . By and , respectively, we mean the classes of analytic functions that satisfy the analytic conditions , and , for introduced and studied by Kanas and Ronning [1]. The class is defined by geometric property that the image of any circular arc centered at is starlike with respect to and the corresponding class is defined by the property that the image of any circular arc centered at is convex. We observe that the definitions are somewhat similar to the ones introduced by Goodman in [2, 3] for uniformly starlike and convex functions, except that in this case the point is fixed. In particular, and , respectively, are the well-known standard classes of convex and starlike functions. Let denote the class of meromorphic functions of the form defined on the punctured unit disk . Denote by the subclass of consisting of the functions of the form A function of the form (4) is in the class of meromorphic starlike of order ( ) denoted by , if and is in the class of meromorphic convex of order ( ) denoted by , if For functions given by (4) and ,？？ we define the Hadamard product or convolution of and by More recently, Purohit and Raina [4] introduced a generalized -Taylor’s formula in fractional -calculus and derived certain -generating functions for -hypergeometric functions. In this work we proceed to derive a generalized differential operator on meromorphic functions in involving these functions and discuss some of their properties. For complex parameters and ？ the -hypergeometric function is defined by with where when . The -shifted factorial is defined for as a product of factors by and in terms of basic analogue of the gamma function It is of interest to note that is the familiar Pochhammer symbol and Now for , , and , the basic hypergeometric function defined in (8) takes the form which converges absolutely in the open unit disk . Corresponding to the function recently for meromorphic functions consisting functions of the form (3), Huda and